On Tiling and Spectral Sets in $\mathbb Z_{p^2}\times\mathbb Z_{p^2}$

Abstract

Let $p$ be a prime number, it is shown that tiling and spectral sets coincide in $\mathbb Z_{p^2}\times\mathbb Z_{p^2}$ by considering equivalently symplectic spectral pairs. The main approach is still to analyze the zero set of the Fourier transform. The zero set of the symplectic Fourier transform differs from the zero set of the usual Fourier transform by an orthogonal rotation, but using the symplectic Fourier transform allows more freedom when applying change of basis. Some auxiliary results concerning tiling sets and spectral sets of sizes $p$ and $p^{2m-1}$ in $\mathbb Z_{p^m}\times\mathbb Z_{p^m}$ are also presented.